Analytic Novikov for topologists (corrected version), by Jonathan Rosenberg

This is a corrected version of an earliar preprint in this archive.

We explain for topologists the ``dictionary'' for understanding the analytic proofs of the Novikov conjecture, and how they relate to the surgery-theoretic proofs. In particular, we try to explain the following points:

Why do the analytic proofs of the Novikov conjecture require the introduction of C*-algebras?

Why do the analytic proofs of the Novikov conjecture all use $K$-theory instead of $L$-theory? Aren't they computing the wrong thing? (In this connection we discuss in detail how the various (algebraic) $L$-theory spectra of a real or complex C*-algebra are related to its TOPOLOGICAL $K$-theory spectrum.)

How can one show that the index map $\mu$ or $\beta$ studied by operator theorists matches up with the assembly map in surgery theory?

Where does ``bounded surgery theory'' appear in the analytic proofs? Can one find a correspondence between the sorts of arguments used by analysts and the controlled surgery arguments used by topologists?


Jonathan Rosenberg <jmr@math.umd.edu>